Hiproofs: A Hierarchical Notion of Proof Tree

Motivated by the concerns of theorem-proving, we generalise the notion of proof tree to that of hierarchical proof tree. Hierarchical trees extend ordinary trees by adding partial order structure to the set of nodes: that allows us to visualise a node as a rectangle in the plane rather than as a point, letting us use the containment relation to express structure additional to that given by a tree. A hierarchical proof tree, or hiproof for short, is a hierarchical tree with nodes labelled by tactics. We motivate the details of our definition by reference to the behaviour of tactics in tactical theorem proving. We characterise the construction of the ordinary proof tree underlying a hierarchical proof tree as a left adjoint. We then analyse the notion of proof refinement with respect to hierarchy, and we give a characterisation of hiproofs that is more directly suited to implementation.